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A B1A003 Total No. of pages:2

Page 1 of 2

Reg. No._______________ Name:__________________________

APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY

THIRD SEMESTER B.TECH DEGREE EXAMINATION, DEC 2016

Course Code: MA201

Course Name: LINEAR ALGEBRA AND COMPLEX ANALYSIS

Max. Marks: 100 Duration:3. Hours

PART A

(Answer any two questions)

1.a Show that � = �� − 3���is harmonic and hence find its harmonic conjugate. (8)

b Find the image of �� − �

� � ≤

�

� under the transformation =

�

� . Also find the fixed points

of the transformation � = �

� (7)

2.a Define an analytic function and prove that an analytic function of constant modulus is

constant. (8)

b Find the linear fractional transformation that maps �� = 0, �� = 1, �� = ∞onto

�� = −1, �� = −�, �� = 1 respectively. (7)

3.a Show that �(�) = ������� − �������� is differentiable everywhere. Find

its derivative. (8)

b Find the image of the lines � = � and � = �, where �&�are constants, under the

transformation � = ����. (7)

PART B

(Answer any two questions)

4.a Evaluate ∫ �� (�) �� �

where � is a straight line from 0 to 1 + 2�. (7)

b Show that ∫ ��

���� =

�

�√�

�

� (8)

5.a Integrate ��

���� counterclockwise around the circle |� − 1 − �| =

�

� by Cauchy’s

Integral Formula. (7)

b Evaluate ∫ ����

������� ��

� where � is |� − 2 − �| = 3.5 by Cauchy’s Residue Theorem

(8)

6.a If �(�) = �

�� find the Taylor series that converges in |� − �| < �and the Laurent’s

series that converges in |� − �| > �. (8)

b Define three types of isolated singularities with an example for each. (7)

Department of Mechanical Engineering, SCMS School of Engineering and Technology.

A B1A003 Total No. of pages:2

Page 2 of 2

PART C

(Answer any two questions)

7.a Solve by Gauss Elimination:

�� − �� + �� = 0,

−�� + �� − �� = 0,

10 �� + 25 �� = 90,

20 �� + 10 �� = 80. (5)

b Find the rank. Also find a basis for the row space and column space for

� 0 1 0 −1 0 −4 0 4 0

� (5)

c Find out what type of conic section the quadratic form

� = 17 �� − 30 �� + 17 �� = 128 represents and transform it to the principal

axes. (10)

8.a Find whether the vectors [1 2−1 3], [2 −13 2]��� [−1 8−9 5] are

linearly dependent. (5)

b Show that the matrix � = � 1 2 2 −2

� is symmetric. Find the spectrum. (5)

c Diagonalise � = � 8 −6 2 −6 7 −4 2 −4 3

� (10)

9. a. Determine whether the matrix

⎣ ⎢ ⎢ ⎡ 1 0 0

0 1 √2

� −1 √2

�

0 1 √2

� 1 √2

� ⎦ ⎥ ⎥ ⎤ is orthogonal? (5)

b. Find the Eigen values and Eigen vectors of � 1 1 2 −1 2 1 0 1 3

� (5)

c. Define a Vector Space with an example. (10)

Department of Mechanical Engineering, SCMS School of Engineering and Technology.

A B3A005 Pages:2

Page 1 of 2

Reg. No._____________ Name:_____________________

APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY

THIRD SEMESTER B.TECH DEGREE EXAMINATION, MARCH 2017

MA 201: LINEAR ALGEBRA AND COMPLEX ANALYSIS

Max. Marks: 100 Duration: 3 Hours

PART A

Answer any 2 questions

1. a. Check whether the following functions are analytic or not. Justify your answer.

i) zzf z (4)

ii) 2zzf

(4)

b. Show that zzf sin is analytic for all z. Find zf (7)

2. a. Show that 323 yyxv is harmonic and find the corresponding analytic function

yxivyxuzf ,, (8)

b. Find the image of 10 x , 1 2

1 y under the mapping zew (7)

3. a. Find the linear fractional transformation that carries �� = −2, �� = 0 and �� = 2

on to the points �� = ∞, �� = 1

4� and �� = 3

8� . Hence find the image of x-axis.(7)

b. Find the image of the rectangular region x , bya under the mapping

zw sin (8)

PART B

Answer any 2 questions

4. a. Evaluate ∫ |�|�� �

where

i) C is the line segment joining -i and i (3)

ii) C is the unit circle in the left of half plane (4)

b. Verify Cauchy’s integral theorem for �� taken over the boundary of the rectangle

with vertices -1, 1, 1+i, -1+i in the counter clockwise sense. (8)

5. a. Find the Laurent’s series expansion of 21

1

z zf

which is convergent in

i) |� − 1| < 2 (4)

ii) |� − 1| > 2 (4)

b. Determine the nature and type of singularities of

i) 2

2

z

e z (3)

Department of Mechanical Engineering, SCMS School of Engineering and Technology.

A B3A005 Pages:2

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ii) � sin (� � )

(4)

6. a. Use residue theorem to evaluate

dz zz

zz

C

1312

52330 2

2

where C is 1z (7)

b. Evaluate

dx x

0 221

1 using residue theorem. (8)

PART C

Answer any 2 questions

7. a. Solve the following by Gauss elimination

y + z – 2w = 0, 2x – 3y – 3z + 6w = 2, 4x + y + z – 2w = 4 (6)

b. Reduce to Echelon form and hence find the rank of the matrix

1502121

5424426

2203

(6)

c. Find a basis for the null space of

402

840

022

(8)

8. a. i) Are the vectors (3 -1 4), (6 7 5) and (9 6 9) linearly dependent or

independent? Justify your answer. (5)

ii) Is all vectors zyx ,, in ℝ� with 04 zxy form a vector space over the field

of real numbers? Give reasons for your answer. (5)

b. i) Find a matrix C such that xCxTQ where

2 331

2 221

2 1 5243 xxxxxxxQ (4)

ii) Obtain the matrix of transformation

y1 = cos θ x1 – sin θ x2, y2 = sin θ x1 + cos θ x2

Prove that it is orthogonal. Obtain the inverse transformation. (6)

9. a. Find the eigenvalues, eigenvectors and bases and dimensions for each Eigen space

of

021

612

322

A

(10)

b. Find out what type of conic section, the quadratic form 128173017 2221 2 1 xxxx

and transform it to principal axes. (10)

Department of Mechanical Engineering, SCMS School of Engineering and Technology.

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